Question

p（x,y)是曲线x^2/25+y^2/16=1上的动点,则(2/5)*x+(3/4)*y的最大值是?

Answer 1

右焦点(3,0),左焦点(-3,0)

设所求点是(m,n)

(m-3)^2+n^2=4[(m+3)^2+n^2]

(m-3)^2-4(m+3)^2=3n^2

(m-3+2m+6)(m-3-2m-6)=3n^2

(m+1)(-m-9)=n^2

代入椭圆

m^2/25+(m+1)(-m-9)/16=1

9m^2+250m+625=0

m=-25,m=-25/9

m=-25,n无解

m=-25/9,n=±8√14/9

所以有两个点

(-25/9,8√14/9),(-25/9,-8√14/9)